Finding potential of a given wavefunction in spherical polar

[sudipmaity]

Homework Statement

The ground state wavefuntion
of a system in spherical polar
coordinates is given by:
Ψ (r,θ, φ)= (A/r) [exp (-ar) -
exp (-br)] where a, b, A are
constants.
i) Determine A as a function
of a and b, so as to normalize
the wavefuntion.
ii) From Schrödinger equation
find V (r) in terms of a and b
iii) From potential behaviour
find the energy eigenvalue if
b=6a in the ground state.

Homework Equations

integral |ψ|^2 dτ=1
Hψ=Eψ
H= -hbar/2m grad^2 + V

The Attempt at a Solution

I integrated | ψ|^2 r^2 dr
sinθ dθ dφ =1
I found A= {1/a-b}[ab* (a
+b) /2π]^(1/2)
After putting the wavefuntion
in time independent
Schrödinger's (Hψ=Eψ)the
calculation is getting pretty
elaborate.Cant figure out how
to find the potential.This is a
University exam question and
each of the three question
carried 2 marks.please help
me calculate the potential.

 

[Simon Bridge]

Please show your working when you applied the Schrödinger equation.

 

[sudipmaity]

let 2m/hbar^2=k
then -1/k grad^2 ψ +Vψ=Eψ
or grad^2 ψ =k (V-E)ψ
given wavefuntion is independent of θ, φ
so in spherical polar grad^2= (1/r^2)δ/δr (r^2 δ/δr)
I am getting (r^2 δ/δr)= r [b exp(-br)-a exp (-ar)] - [exp (-ar)-exp (-br)]
next differenting the above again w.r.t r and multipying with 1/r^2
grad^2 ψ= ( 1/r) [a^2 exp (-ar)- b^2 exp (-br)]
'A' depends on a and b which are constants .So I guess A will cancel out.
So now my TISE looks like
( 1/r) [a^2 exp (-ar)- b^2 exp (-br)]=k (V-E) ( 1/r) [exp (-ar)- exp (-br)]
What should I do after this??

 

[sudipmaity]

Can somebody at all confirm whether it is possible to find the potential?

 

[sudipmaity]

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